Contrast constrained aerothermal radiation correction method

ABSTRACT

Disclosed in the present invention is a contrast constrained aerothermal radiation correction method. By analyzing features of images at different intensities of aerothermal radiation, it has been discovered that the stronger the aerothermal radiation effect is, the smaller the image contrast becomes, and when thermal radiation correction is performed using a gradient fitting algorithm, it has been discovered that time consumption thereof grows exponentially with an increase in a degree of a fitting surface and with an increase in an image size. The present invention can rapidly and effectively restore an aerothermal radiation image, remarkably improving a signal to noise ratio and quality of the image.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a National Stage Appl. filed under 35 USC 371 ofInternational Patent Application No. PCT/CN2016/079129 with aninternational filing date of Apr. 13, 2016, designating the UnitedStates, and further claims foreign priority benefits to Chinese PatentApplication No. 201510988503.9 filed Dec. 24, 2015. Inquiries from thepublic to applicants or assignees concerning this document or therelated applications should be directed to: Matthias Scholl P.C., Attn.:Dr. Matthias Scholl Esq., 245 First Street, 18th Floor, Cambridge, Mass.02142.

TECHNICAL FIELD

The present disclosure relates to the field of an interdisciplinaryscience combining aero-optics, information processing and aerospacetechnology, and more particularly to a contrast-constrained aerothermalradiation correction method which can be applied to image preprocessingfor remote sensing, detection, navigation and guidance of high-speedaircraft.

BACKGROUND OF THE INVENTION

Aero-optics is an interdisciplinary science combining aerodynamics andoptics. For high-speed aircraft flying in the atmosphere, severeaero-optical effect will occur, which affects the imaging quality of anoptical imaging detection system. Therefore, the aero-optical effect,along with its correction method, is an important research direction,and one of the main technical problems that restrict the development andapplication of high-speed aircrafts.

For high-speed aircraft with an optical imaging detection system flyingin the atmosphere, a complex airflow field is formed by the interactionbetween an optical window and the airflow. Due to air viscosity, theairflow in contact with the surface of the optical window will beretarded, so that the airflow velocity decreases and a boundary layer isformed near the surface of the optical window. Within the boundarylayer, the airflow layers with a large velocity gradient will havestrong friction, converting kinetic energy of the airflow irreversiblyinto heat, and leading to rise of the temperature in the wall of theoptical window. The high-temperature airflow will continuously transferheat to the low-temperature walls, causing strong aerothermal heating.The optical window is aerothermal-heated and hence in a severeaerothermal environment; as a result, it produces thermal radiationnoise, reduces signal-to-noise ratio and degrades image quality of theoptoelectronic detection system.

The greater the flight speed, the more severe the aerothermal heating onthe surface of the aircraft. The irradiance of the airflow outside ofthe optical window and the irradiance of the optical window aresuperimposed on the irradiance of background; as a result, an imagingsensor will enter a non-linear operation range or saturate, causing lossof effective information of scenes and reduction of signal-to-noiseratio and signal-to-clutter ratio, thus degrading the detectionperformance. Therefore, it is necessary to perform aerothermal radiationcorrection on the images acquired by the imaging sensor, in order toimprove the image quality. As degradation model of aerothermal radiationis unknown and randomly changed and the degraded images contain othernoise, these increase the difficulty of image restoration or correction.In addition, for specific applications in high-speed aircrafts,especially hypersonic aircrafts, the high-frame-rate characteristic oftheir imaging systems demands that a correction algorithm have extremelyhigh real-time performance.

SUMMARY OF THE INVENTION

In view of the above-described problems, it is one objective of theinvention to provide a contrast-constrained aerothermal radiationcorrection method configured to solve the defects of the conventionalaerothermal radiation correction methods, i.e., poor results in handlingstrong thermal radiation effect and low correction efficiency inprocessing large-size images; the method provided by the presentdisclosure can be used for aerothermal radiation correction on imagesacquired during remote sensing, detection, navigation and guidance of ahigh-speed aircraft.

To achieve the above objective, in accordance with one embodiment of theinvention, there is provided an aerothermal radiation correction method,the method comprising:

-   -   (1) filtering out noise and details in an original aerothermal        radiation image Z, thus obtaining a filtered image {circumflex        over (Z)}, to overcome the adverse effects of noise in thermal        radiation field estimation process;    -   (2) through estimation of the filtered image {circumflex over        (Z)}, obtaining an aerothermal radiation field B₁ of the        original aerothermal radiation image Z, and further obtaining an        initially corrected image S₁=Z−B₁;    -   (3) solving a central region of the aerothermal radiation field        B₁, and according to the central region of the aerothermal        radiation field B₁, dividing the original aerothermal radiation        image Z and the initially corrected image S₁ into        correspondingly equal-sized image-blocks;    -   (4) calculating the contrast values of the image-blocks of the        original aerothermal radiation image Z and the contrast values        of the image-blocks of the initially corrected image S₁,        respectively, thus obtaining the variation of the contrast        values of the image-blocks of the original aerothermal radiation        image Z relative to the corresponding image-blocks of the        initially corrected image S₁.    -   (5) comparing the variation of the contrast values of the        image-blocks corresponding to the central region of radiation        and the variation of the contrast values of the image-blocks        corresponding to non-central regions of radiation, if the        difference is less than or equal to a predetermined threshold        value, then taking the initially corrected image S₁ as the final        correction result, otherwise sequentially proceeding to step        (6); and    -   (6) obtaining an image        ={circumflex over (Z)}−B₁ from the filtered image {circumflex        over (Z)} and the aerothermal radiation field B₁, and taking the        portion of the image        corresponding to the radiation central region of the aerothermal        radiation field B₁ as a new filtered image {circumflex over        (Z)}, and through estimation of the new filtered image        {circumflex over (Z)}, obtaining a residual aerothermal        radiation field B₂ at the radiation central region of the        initially corrected image S₁, and further obtaining a        secondarily corrected image S₂=S₁−B₂.

In a class of this embodiment, the above method further comprises a step(7) as follows: performing weighted processing to the edges of theradiation central region of the secondarily corrected image S₂ and theedges of the radiation central region of the initially corrected imageS₁, to eliminate the edge effect caused by block division, to achievehigher quality of the images.

In a class of this embodiment, in step (3), taking the radiation centralregion of the aerothermal radiation field B₁ as the center, and dividingthe original aerothermal radiation image Z and the initially correctedimage S₁ into correspondingly equal-sized image-blocks, in such a waythat the image-blocks corresponding to the radiation center regions ofthe aerothermal radiation field B₁ and of the original aerothermalradiation image Z are located at the center of all the image-blocks ofthe original aerothermal radiation image Z, and that the image-blockscorresponding to the radiation center regions of the aerothermalradiation field B₁ and of the initially corrected image S₁ are locatedat the center of all the image-blocks of the initially corrected imageS₁.

In general, the above technical solution contemplated by the presentdisclosure has the following advantageous effects as compared with theprior art: by analyzing features of aerothermal radiation images withdifferent intensities, it is found that, for an image, the stronger theaerothermal radiation effect, the smaller the contrast. When thermalradiation correction is carried out by using a gradient fittingalgorithm, it is found that, the time consumption increasesexponentially with the increase of the order of the fittedcurved-surface and the image size; the present disclosure can achievequick and effective aerothermal radiation image restoration, therebysignificantly improving signal-to-noise ratio and quality of images.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph showing image contrast value varies with intensity ofaerothermal radiation, by adding a curved-surface withgradually-increased irradiance to an input image and calculatingvariance of contrast values;

FIG. 2 is a graph showing time consumption varies with fitting order K,in a correction method of gradient least-squares fitting;

FIG. 3 is a graph showing time consumption varies with image size, in acorrection method of gradient least-squares fitting;

FIG. 4 is a flowchart of the contrast-constrained aerothermal radiationcorrection method according to an embodiment of the present disclosure;

FIG. 5 shows the filtering effect of weighted least-squares (WLS),where, (a) is an original image, (b) is a filtered image;

FIG. 6 shows an example of the contrast-constrained aerothermalradiation correction method, where, (a) is a reference image; (b) is anaerothermal-radiation degraded image; (c) is the result of WLS filteringof (b); (d) is the result of initial correction of (b), through gradientfitting of an aerothermal radiation curved-surface, in the case of K=2;(e) is a schematic diagram illustrating the image-blocks division to thefitted aerothermal radiation field of (d) in the case of K=2 and theinitially corrected result; (f) is the result of secondary correction ofthe central region of (e); (g) is the result of edge fusion of (e); (h)is a graph of a column of pixels taken from (g), for verification of theresult and effect of aerothermal radiation correction;

FIG. 7 shows comparison of the contrast-constrained aerothermalradiation correction method and a unconstrained aerothermal radiationcorrection method, where, (a) is the correction result in the case ofK=9; (b) is the result of contrast-constrained correction; (c) is graphof a column of pixels taken respectively from a reference image, anunconstrained-corrected image, and a contrast-constrained-correctedimage, showing the respective effects; and

FIG. 8 shows the respective effects of electric-arc-wind-tunnelexperiments, where, (a) is an electric-arc-wind-tunnel aerothermalradiation image; (b) is the result of an unconstrained correction; (c)is the result of contrast-constrained correction; (d) is a graph of acolumn of pixels taken from the respective results, showing therespective effects.

DETAILED DESCRIPTION OF THE EMBODIMENTS

For further illustrating the invention, experiments detailing acontrast-constrained aerothermal radiation correction method aredescribed hereinbelow combined with the drawings. It should be notedthat the following examples are intended to describe but not to limitthe invention.

The present disclosure is based on the following three characteristicsof thermal radiation correction: (1) image contrast value decreases withgradual increase of aerothermal radiation intensity, as shown in FIG. 1;(2) when thermal radiation correction is carried out by using gradientfitting, the time consumption increases exponentially with increase offitting order K, as shown in FIG. 2; (3) when thermal radiationcorrection is carried out by using gradient fitting, the correction timeconsumption increases approximately in exponential form with increase ofimage size, as shown in FIG. 3; therefore, a contrast-constrainedaerothermal radiation correction method is proposed.

As shown in FIG. 4, the contrast-constrained aerothermal radiationcorrection method according to an embodiment of the present disclosurecomprises the following steps:

(1) Due to the low-frequency characteristics of an aerothermal radiationfield, using an image-smoothing algorithm based on WLS to filter outnoise and details in an original aerothermal radiation image Z, thusobtaining a filtered image {circumflex over (Z)}, to overcome theadverse effects of noise in thermal radiation field estimation process;

Specifically, the filtered image {circumflex over (Z)} is expressed as:

{circumflex over (Z)}=(I+λL)⁻¹ Z,

where, I is a unit matrix; λ is a smooth parameter, the larger λ, thesmoother the filtered image, and in this embodiment, it is set to 0.002;L=D_(x) ^(T)W_(x)D_(x)+D_(y) ^(T)W_(y)D_(y), D_(x) and D_(y) are forwarddifference operators in x direction and y direction of the imagecoordinate system, respectively; W_(x) and W_(y) are weighting-factordiagonal matrixes in x direction and y direction of the image coordinatesystem, respectively.

Specifically, the value of the diagonal element in the i-th row of W_(x)is

$\left( {{{\frac{\partial }{\partial x}(i)}}^{\alpha} + ɛ} \right)^{- 1},$

the value of the diagonal element in the i-th row of W_(y) is

$\left( {{{\frac{\partial }{\partial y}(i)}}^{\alpha} + ɛ} \right)^{- 1},$

where, l is the logarithmic transformation of an input image Z;

$\frac{\partial }{\partial x}\mspace{14mu} {and}\mspace{14mu} \frac{\partial }{\partial y}$

represent gradients in x direction and y direction, respectively; α is aconstant for control of a filter's sensitivity to the gradient of anoriginal image, the larger α, the more sensitive to the gradient of theoriginal image, and the stronger edge-retention of the image afterfiltering, and in this embodiment, it is set to 1.1; ε is a relativelysmall constant for preventing denominator from being 0, and in thisembodiment, it is set to 0.00001.

FIG. 5 shows the WLS filtering effect, where, (a) in FIG. 5 is anoriginal image, (b) in FIG. 5 is a filtered image; it is apparent fromthe figure, after the WLS filtering processing, most of the details ofthe image are filtered out.

Based on this feature of WLS, the filtering process is performed onaerothermal-radiation degraded images; (a) in FIG. 6 is an originalinfrared image, (b) in FIG. 6 is an aerothermal-radiation degradedimage; (c) in FIG. 6 is the image after the WLS filtering processing,from which high-frequency components (noise and details) are filteredout, making estimation of a low-frequency aerothermal radiation field insubsequent steps more accurate.

(2) Based on the feature that an aerothermal radiation field B can befitted by a K-order two-dimensional polynomial, using a least-squaresmethod for estimation of the filtered image {circumflex over (Z)}, thusobtaining the aerothermal radiation field B of the original aerothermalradiation image Z; in this step, the aerothermal radiation fieldobtained through estimation with the setting K=2 is denoted as B₁, andin the subsequent step (6), the initially corrected residual aerothermalradiation field obtained through estimation is denoted as B₂.

Further, from the obtained aerothermal radiation field B₁, an initialcorrected image S₁=Z−B₁ is obtained;

${B\left( {x,y} \right)} = {{\sum\limits_{t = 0}^{K}{\sum\limits_{s = 0}^{K - t}{a_{t,s}x^{t}y^{s}}}} = {Ca}}$

where, (x, y) are coordinates of a pixel; a is polynomial coefficient; Cis a constant matrix for substituting x and y into the above equation; Kis the order of the two-dimensional polynomial.

argmin∥∇{circumflex over (Z)}−∇B∥ ₂ ²

A least-squares method is employed for estimation, and when the L2 normof the difference between the gradient ∇{circumflex over (Z)} of thefiltered image {circumflex over (Z)} and the gradient ∇B of theaerothermal radiation field to be estimated arrives at a minimumsolution, that is the solved aerothermal radiation field B, as shown bythe above formula.

In the solving process of this approach, the time consumption has thetrend of exponential growth with the increase of K, as shown in FIG. 2;therefore, in the first use of the above approach, the aerothermalradiation field B₁ is solved with the setting K=2, thus obtaining theinitial corrected image S₁, as shown in FIG. 6(d), and by subtractingthe radiation field B₁ from the filtered image, an image

is obtained,

is used as a filtered image and as an input image in subsequent stepsfor secondary estimation of radiation field. It can be seen from theinitially corrected image that, most of the radiation field of the imagehas been filtered out, and only the radiation central region remainsrelatively strong radiation field; therefore, the subsequent steps focuson estimation and removal of the thermal radiation field at theradiation central region.

(3) Solving the radiation central region of the aerothermal radiationfield B₁, wherein the radiation central region corresponds to the regionhaving relatively large gray-scale values in the thermal radiationimage; and according to the radiation central region of the aerothermalradiation field B₁, dividing the original aerothermal radiation image Zand the initially corrected image S₁ into correspondingly equal-sizedimage-blocks;

For example, by utilizing the solved radiation central region,block-division processing is performed on the input aerothermalradiation image (as shown in (b) in FIG. 6) and the initially correctedimage (as shown in (d) in FIG. 6), thus, with the radiation centralregion as the center, the images are each divided into 9 image-blocks,and the result is as shown in (e) in FIG. 6.

(4) Calculating the contrast values of the image-blocks of the originalaerothermal radiation image Z and the contrast values of theimage-blocks of the initially corrected image S₁, respectively, thusobtaining the variation of the contrast values of the image-blocks ofthe original aerothermal radiation image Z relative to the correspondingimage-blocks of the initially corrected image S₁;

For example, a method for calculating the contrast value of animage-block is as follows:

Calculating the sum of the squares of the difference between thegray-scale value of each pixel and the gray-scale values of the fouradjacent pixels, then dividing the sum by the total pixel number of theimage block.

Ctr=Σ_(δ)δ(i,j)² P _(δ)(i,j)

where, δ(i, j) is the gray-scale difference between adjacent pixels;P_(δ)(i, j) is the distribution probability of a pixel with gray-scaledifference between adjacent pixels being δ; Ctr is the contrast value ofthe image.

With the above contrast-value calculation formula, the contrast valuesof the image-blocks of the original aerothermal radiation image Z andthe contrast values of the image-blocks of the initially corrected imageS₁ are calculated respectively, as shown in Table 1 and Table 2. Table 1shows the contrast values of the respective image-blocks of the originalaerothermal radiation image Z, Table 2 shows the contrast values of therespective image-blocks of the initially corrected image S₁, and Table 3shows the difference between the contrast values of the correspondingimage-blocks of both the original aerothermal radiation image Z and theinitially corrected image S₁.

TABLE 1 41.5 26.7 83.2 43.7 18.0 70.3 100.3 37.1 22.8

TABLE 2 126.0 169.8 337.4 220.9 44.6 271.5 237.3 126.2 103.8

TABLE 3 84.4 143.1 254.2 177.1 26.5 201.1 137.0 89.1 80.9

(5) Based on the feature that, for an aerothermal radiation image, thestronger the radiation, the smaller the contrast, comparing andanalyzing the variation of the contrast values of the image-blocks ofthe original aerothermal radiation image Z relative to the correspondingimage-blocks of the initially corrected image S₁, which is obtained instep (4); if the variation of the contrast values at the central regionof radiation is significantly less than the variation of the contrastvalues at non-central regions of radiation, that is, the difference isgreater than a predetermined threshold value T, which indicates that:the radiation intensity of the original aerothermal radiation image Z isquite high, although the effect of correction for non-central regions ofradiation has reached to a relatively high level, the central region ofradiation still remains relatively strong aerothermal radiation noise,hence, it is necessary to perform secondary correction to the centralregion of radiation, thus sequentially proceeding to step (6);

Contrarily, if the variation of the contrast values at the centralregion of radiation is slightly different from the variation of thecontrast values at non-central regions of radiation, that is, thedifference is less than or equal to a predetermined threshold value T,which indicates that: the radiation intensity of the originalaerothermal radiation image Z is relatively low, so the image has beenwell corrected, thus taking the initially corrected image S₁ as thefinal correction result;

It can be clearly seen from Table 3 that, the restoration of thecontrast value at the central region of radiation is significantly lowerthan that at the peripheral regions, which indicates that the centralregion of radiation still remains relatively strong aerothermalradiation noise, so it is necessary to perform secondary correction tothe central region of radiation.

(6) Obtaining an image

₁={circumflex over (Z)}−B from the filtered image {circumflex over (Z)}resulted in step (1) and the aerothermal radiation field B₁ resulted instep (2), and taking the portion of the image

₁ corresponding to the radiation central region of the aerothermalradiation field B₁ as a new filtered image Z, and through estimation ofthe new filtered image {circumflex over (Z)}, obtaining a residualaerothermal radiation field B₂ at the radiation central region of theinitially corrected image S₁, and further obtaining a secondarilycorrected image S₂=S₁−B₂;

In the fitting process herein, since only the relatively small region atthe center of the image is subjected to the fitting process, it ispossible to use a higher order K—compared with the case in step (2)—forthe fitting process, without causing a high cost of time consumption; inthis embodiment, K=5. By subtracting the estimated aerothermal radiationfield at the central region from the central region of the initiallycorrected image, a secondary corrected image S₂ is obtained, as shown in(f) in FIG. 6.

(7) Due to the edge effect inevitably brought by block divisionprocessing of images, in this step, performing weighted processing tothe edges of the radiation central region of the secondarily correctedimage S₂ and the edges of the radiation central region of the initiallycorrected image S₁, to eliminate the edge effect caused by blockdivision, to achieve higher quality of the images.

It can be seen from (f) in FIG. 6 that, the aerothermal radiation noisehas been completely removed, but the edge effect brought by blockdivision still remains; the image after weighted processing is as shownin (g) in FIG. 6. (h) in FIG. 6 shows a graph of a column of pixelstaken from the original image, the aerothermal radiation image and thecontrast-constrained corrected image, respectively, and it can be seenfrom the graph that, after processing according to the method of thepresent disclosure, the aerothermal radiation effect in the image hasbeen well corrected.

FIG. 7 shows the result of aerothermal-radiation-field gradient fittingby directly using K=9 ((a) in FIG. 7) and the result ofcontrast-constrained secondary correction ((b) in FIG. 7); it isapparent from the figure that, directly using high-order fitting leadsto generation of a relatively strong halation ring at the central regionof aerothermal radiation, thus affecting image quality, whereascontrast-constrained secondary correction overcomes this problem; (c) inFIG. 7 is a graph of a column of pixels taken from an original image, anaerothermal radiation image, a corrected image by using high-order (K=9)two-dimensional polynomial fitting and a contrast-constrained-correctedimage, respectively, showing that the method of the present disclosurereduces correction time consumption while guaranteeing correctioneffect.

Below, Table 4 shows comparison of the respective parameters, such astime consumption and PSNR (peak-signal-to-noise ratio), of a directcorrection method and the contrast-constrained correction method incorrection of the same aerothermal radiation image, and it can beclearly seen from the table that, the contrast-constrained correctionmethod greatly reduces correction time consumption, with slightlyincrease of PSNR.

TABLE 4 Images T(s) PSNR(dB) Aerothermal radiation image — 12.34Directly corrected image 0.7393 19.23 Contrast-constrained-correctedimage 0.4322 22.85

FIG. 8 is an example for correcting an electric-arc-wind-tunnel image,where, (a) in FIG. 8 is a wind-tunnel aerothermal radiation image, (b)in FIG. 8 is the corrected image by using high-order (K=9)two-dimensional polynomial fitting, (c) in FIG. 8 is thecontrast-constrained-corrected image, (d) in FIG. 8 is a graph of acolumn of pixels taken from the electric-arc-wind-tunnel aerothermalradiation image, the corrected image by using high-order (K=9)two-dimensional polynomial fitting and thecontrast-constrained-corrected image, respectively.

Unless otherwise indicated, the numerical ranges involved in theinvention include the end values. While particular embodiments of theinvention have been shown and described, it will be obvious to thoseskilled in the art that changes and modifications may be made withoutdeparting from the invention in its broader aspects, and therefore, theaim in the appended claims is to cover all such changes andmodifications as fall within the true spirit and scope of the invention.

The invention claimed is:
 1. An aerothermal radiation correction method,the method comprising: (1) filtering out noise and details in anoriginal aerothermal radiation image Z, thus obtaining a filtered image{circumflex over (Z)}, to overcome the adverse effects of noise inthermal radiation field estimation process; (2) through estimation ofthe filtered image {circumflex over (Z)}, obtaining an aerothermalradiation field B₁ of the original aerothermal radiation image Z, andfurther obtaining an initially corrected image S₁=Z−B₁; (3) solving acentral region of the aerothermal radiation field B₁, and according tothe central region of the aerothermal radiation field B₁, dividing theoriginal aerothermal radiation image Z and the initially corrected imageS₁ into correspondingly equal-sized image-blocks; (4) calculating thecontrast values of the image-blocks of the original aerothermalradiation image Z and the contrast values of the image-blocks of theinitially corrected image S₁, respectively, thus obtaining the variationof the contrast values of the image-blocks of the original aerothermalradiation image Z relative to the corresponding image-blocks of theinitially corrected image S₁; (5) comparing the variation of thecontrast values of the image-blocks corresponding to the central regionof radiation and the variation of the contrast values of theimage-blocks corresponding to non-central regions of radiation, if thedifference is less than or equal to a predetermined threshold value,then taking the initially corrected image S₁ as the final correctionresult, otherwise sequentially proceeding to step (6); and (6) obtainingan image

₁={circumflex over (Z)}−B₁ from the filtered image {circumflex over (Z)}and the aerothermal radiation field B₁ and taking the portion of theimage

corresponding to the radiation central region of the aerothermalradiation field B₁ as a new filtered image {circumflex over (Z)}, andthrough estimation of the new filtered image {circumflex over (Z)},obtaining a residual aerothermal radiation field B₂ at the radiationcentral region of the initially corrected image S₁, and furtherobtaining a secondarily corrected image S₂=S₁−B₂.
 2. The method of claim1, further comprising a step (7) as follows: performing weightedprocessing to the edges of the radiation central region of thesecondarily corrected image S₂ and the edges of the radiation centralregion of the initially corrected image S₁, to eliminate the edge effectcaused by block division, to achieve higher quality of the images. 3.The method of claim 1 or 2, wherein in step (3), taking the radiationcentral region of the aerothermal radiation field B₁ as the center, anddividing the original aerothermal radiation image Z and the initiallycorrected image S₁ into correspondingly equal-sized image-blocks, insuch a way that the image-blocks corresponding to the radiation centerregions of the aerothermal radiation field B₁ and of the originalaerothermal radiation image Z are located at the center of all theimage-blocks of the original aerothermal radiation image Z, and that theimage-blocks corresponding to the radiation center regions of theaerothermal radiation field B₁ and of the initially corrected image S₁are located at the center of all the image-blocks of the initiallycorrected image S₁.